Work-modulated queues with applications to storage processes

Abstract

We study two FIFO single-server queueing models in which both the arrival and service processes are modulated by the amount of work in the system. In the first model, the nth customer's service time, Sn, depends upon their delay, Dn, in a general Markovian way and the arrival process is a non-stationary Poisson process (NSPP) modulated by work, that is, with an intensity that is a general deterministic function g of work in system V(t). Some examples are provided. In our second model, the arrivals once again form a work-modulated NSPP, but, each customer brings a job consisting of an amount of work to be processed that is i.i.d. and the service rate is a general deterministic function r of work. This model can be viewed as a storage (dam) model (Brockwell et al. (1982)), but, unlike previous related literature, (where the input is assumed work-independent and stationary), we allow a work-modulated NSPP. Our approach involves an elementary use of Foster's criterion (via Tweedie (1976)) and in addition to obtaining new results, we obtain new and simplified proofs of stability for some known models. Using further criteria of Tweedie, we establish sufficient conditions for the steady-state distribution of customer delay and sojourn time to have finite moments.